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A204270 a(n) = tau(n)*Pell(n), where tau(n) = A000005(n), the number of divisors of n. 23
1, 4, 10, 36, 58, 280, 338, 1632, 2955, 9512, 11482, 83160, 66922, 323128, 780100, 2354160, 2273378, 16465260, 13250218, 95966568, 154455860, 372889432, 450117362, 4346717760, 3935214363, 12667263848, 30581480180, 110745336312, 89120964298 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Compare g.f. to the Lambert series identity: Sum_{n>=1} x^n/(1-x^n) = Sum_{n>=1} tau(n)*x^n.
Related identities:
(1) Sum_{n>=1} n^k*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_{k}(n)*Pell(n)*x^n for k>=0.
(2) Sum_{n>=1} phi(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} n*Pell(n)*x^n.
(3) Sum_{n>=1} moebius(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = x.
(4) Sum_{n>=1} lambda(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} Pell(n^2)*x^(n^2).
LINKS
FORMULA
G.f.: Sum_{n>=1} Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} tau(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.
EXAMPLE
G.f.: A(x) = 1 + 4*x + 10*x^2 + 36*x^3 + 58*x^4 + 280*x^5 + 338*x^6 +...
where A(x) = x/(1-2*x-x^2) + 2*x^2/(1-6*x^2+x^4) + 5*x^3/(1-14*x^3-x^6) + 12*x^4/(1-34*x^4+x^8) + 29*x^5/(1-82*x^5-x^10) + 70*x^6/(1-198*x^6+x^12) +...+ Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...
MATHEMATICA
Table[DivisorSigma[0, n] Fibonacci[n, 2], {n, 1, 50}] (* G. C. Greubel, Jan 05 2018 *)
PROG
(PARI) /* Subroutines used in PARI programs below: */
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}
(PARI) {a(n)=sigma(n, 0)*Pell(n)}
(PARI) {a(n)=polcoeff(sum(m=1, n, Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
CROSSREFS
Sequence in context: A152916 A222506 A108596 * A287220 A220205 A196880
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 14 2012
STATUS
approved

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)